Civil Engineering Reference
In-Depth Information
shaded area in the figure) may be significant. For the case depicted, a large num-
ber of very small square elements best approximates the geometry.
At this point, the astute reader may think, Why not use triangular and rec-
tangular elements in the same mesh to improve the model? Indeed, a combina-
tion of the element types can be used to improve the geometric accuracy of the
model. The shaded areas of Figure 6.20c could be modeled by three-node tri-
angular elements. Such combination of element types may not be the best in
terms of solution accuracy since the rectangular element and the triangular ele-
ment have, by necessity, different order polynomial representations of the field
variable. The field variable
is
continuous across such element boundaries; this is
guaranteed by the finite element formulation. However, conditions on derivatives
of the field variable for the two element types are quite different. On a curved
boundary such as that shown, the triangular element used to fill the “gaps” left by
the rectangular elements may also have adverse aspect ratio characteristics.
Now examine Figure 6.20d, which shows the same area meshed with rectan-
gular elements and a new element applied near the periphery of the domain. The
new element has four nodes, straight sides, but is not rectangular. (Please note
that the mesh shown is intentionally coarse for purposes of illustration.) The new
element is known as a general two-dimensional
quadrilateral element
and is seen
to mesh ideally with the rectangular element as well as approximate the curved
boundary, just like the triangular element. The four-node quadrilateral element is
derived from the four-node rectangular element (known as the
parent
element)
element via a
mapping
process. Figure 6.21 shows the parent element and its
natural (
r
,
s
) coordinates and the quadrilateral element in a global Cartesian
coordinate system. The geometry of the quadrilateral element is described by
4
x
=
G
i
(
x
,
y
)
x
i
(6.77)
i
=
1
4
y
=
G
i
(
x
,
y
)
y
i
(6.78)
i
=
1
where the
G
i
(
x
,
y
)
can be considered as
geometric
interpolation functions, and
each such function is associated with a particular node of the quadrilateral
4 (
x
4
,
y
4
)
4 (
1, 1)
s
3 (1, 1)
3 (
x
3
,
y
3
)
r
y
1 (
1,
1)
1 (
x
1
,
y
1
)
2 (1,
1)
x
2 (
x
2
,
y
2
)
Figure 6.21
Mapping of a parent element into an isoparametric
element. A rectangle is shown for example.
